# Inverter Heat Pumps as a Variable Load for Off-Grid Solar-Powered Systems

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model of the Power Supply System

#### 2.1.1. Battery Energy Storage

#### 2.1.2. Heat Pump

_{min}and f

_{max}. Those are determined by the type of the compressor; continuous operation beyond these frequencies can lead to its rapid failure. It is assumed that compressor power is directly proportional to AC frequency. For heat pump COP dependence on frequency, see Section 2.3. Therefore, HP actual power consumption (P

_{HP}, W) is related to its rated electric power (P

_{HP_Rated}, W) and frequency (f

_{Rated}, Hz) as [18]:

#### 2.2. Weather Data

#### 2.3. Load Analysis

^{2}× K)) is calculated according to an empirical equation (depending on the wind speed (V, m/s)) [32]:

_{amb}, °C) is calculated by [32]:

_{R}, W

_{e}) consumed by the HP compressor for soil thermal stabilisation of a certain area (S, m

^{2}) can be expressed as [32]:

^{2}. Power characteristics of the system will actually vary proportionally to the stabilised area: both HPs and PVs have rather small power steps between the different models. The distribution of HP electric power demanded for thermal stabilisation according to (6) is shown in Figure 4.

_{e}. Consequently, the related power of the HP (P

_{HP_Rated}, W

_{e}) should be sufficient to cover the corresponding annual demand for refrigeration capacity.

#### 2.4. Considered Layouts and Loads

_{min}) and the maximum permissible power (f

_{max}). The aggregation uses several low-power HPs with a total rated capacity equivalent to a single HP. The adaptive load mode uses the possibility of energy storage in the soil to consume all available energy generated by PV regardless of the actual cooling demand. Non-adaptive mode of operation, on the contrary, is aimed primarily at the consumption of minimum required power (according to the actual cooling demand; Figure 4). However, in the calculation, when considering non-adaptive modes of operation, the cold storage in the soil must still be taken into account. Hence, it is impossible to unambiguously consider the «Start–Stop» operation mode in both adaptive and non-adaptive modes for their subsequent comparison. Therefore, the «Start–Stop» mode will be considered only under consideration of the adaptive load.

#### 2.5. Calculation Algorithm

_{C}, J) (during the season of thermal stabilisation system operation) to the total required electric energy (Figure 4) [18]:

- PV energy output.
- The amount of energy consumed by HP (E
_{C}’(t) = P_{R}(t) $\times $ Δt) (4), depending on the load mode, the amount of energy generated, and energy stored in BES. - For BES, the difference between energy generated and energy consumed. In this case, no more energy can be stored in BES than maximum possible energy during the calculated time interval.
- The energy capacity of BES and its SoC. SoC is restricted by maximum and minimum values.

_{C}can be calculated as the sum of E

_{C}’(t). Accordingly, it is possible to calculate RPS (8). Also, based on the analysis of SoC profile, the lifetime of BES can be estimated. This process is based on finding SoC throughout the calculated time interval—each BES discharge cycle leads to its lifetime decrease by a certain amount (which can be estimated by Equations (1)–(3)).

#### 2.6. The Equipment

_{4}BES characteristics were used in the calculations. Lithium batteries have longer life span and also allow greater depths of discharge compared to more usual AGM or GEL batteries. Accordingly, they do not require frequent replacement. These facts contribute to the conclusion that lithium batteries are more cost-effective for stand-alone systems [35] despite higher cost at initial purchase.

_{4}is 20%. The information about the dependence of service life on the depth of discharge and the battery capacity dependence on temperature was taken from the «SunStonePower» manufacturer’s website (see Supplementary Materials Figures S1 and S2) and converted into dependencies convenient for analysis; polynomial coefficients (according to Equations (1) and (2)) are presented in Table 4 and Table 5. The plots of temperature correction coefficients for BES capacity and lifetime and the plot for BES lifetime estimation are shown in Figure 5, Figure 6 and Figure 7, respectively. It is assumed that BES cost depends linearly on the capacity.

## 3. Results

#### 3.1. «Classic» (Configurations Numbered 1–6)

#### 3.2. «Adaptive» (Configurations Numbered 7–14)

## 4. Discussion

## 5. Conclusions

^{2}area in northern Norway, the best configuration of an active thermal stabilisation system is a system with two inverter HPs (installed power of about 3.25 kWe each) and a power supply system based on PV (total installed power capacity of about 47 kW without BES). However, for real systems, it is recommended to install a slightly more expensive system with a power supply system based on PV (43.5 kW) and BES (4.35 kW × h) to increase the HPs’ operation reliability in cloudy weather conditions.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

AC | Alternating current |

BES | Battery Energy Storage |

BOS | Balance of system equipment (inverter and maximum power point tracker) |

CAPEX | CAPital EXpenditure, $ |

COP | Coefficient of performance of the heat pump? heating to electric power ratio |

E_{B}^{inst} | Battery energy storage installed capacity, J |

E_{C} | Heat pump consumed energy for operation season, J |

E’_{C} | Heat pump consumed energy for a certain time period, J |

f | Heat pump compressor AC frequency, Hz |

EER | Energy efficiency rate of the heat pump, cooling to electric power ratio |

f_{max} | Minimum limiting heat pump compressor AC frequency, Hz |

f_{min} | Maximum limiting heat pump compressor AC frequency, Hz |

f_{Rated} | Rated heat pump compressor AC frequency, Hz |

HP | Heat Pump |

GHI | Global Horisontal Irradiance, W/m^{2} |

KT | Service life temperature coefficient |

LT | Number of discharge cycles |

PV | Photo-Voltaic module |

P_{W} | WT instaled capacity, W |

RES | Renewable Energy Sources |

RPS | Reliability of Power Supply |

SoC | BES State of Charge |

t_{amb} | Ambient air temperature, °C |

TCO | Total Cost of Ownership, $ |

TMY | Typical Meteorological Year |

ULT | Portion of a service life consumed as a result of discharge process |

P_{E} | Total installed PV power capacity, W |

P_{HP} | Heat pump power consumption, W_{e} |

P_{HP_Rated} | Heat pump rated electrical power, W_{e} |

P_{R} | Heat pump electrical power necessary for thermal stabilisation, We |

PVGIS | Photovoltaic Geographic Information System |

Δt | Certain time period, s |

η | Complex efficiency factor of the controller devices |

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**Figure 8.**Comparison of the classic and the adaptive load approaches in terms of CAPEX (

**a**) and TCO (

**b**).

**Figure 9.**Dependence of CAPEX (

**a**) and TCO (

**b**) on thermal stabilisation system performance for layouts 1–3 (for systems with aggregation, the number of HPs is given in parentheses).

**Figure 10.**Dependence of CAPEX (

**a**) and TCO (

**b**) on thermal stabilisation system performance for layouts 4–6 (for systems with aggregation, the number of HPs is given in parentheses).

**Figure 11.**Dependence of CAPEX (

**a**) and TCO (

**b**) on thermal stabilisation system performance for layouts 7–10 (for systems with aggregation, the number of HPs is given in parentheses).

**Figure 12.**Dependence of CAPEX (

**a**) and TCO (

**b**) on thermal stabilisation system performance for layouts 11–14 (for systems with aggregation, the number of HPs is given in parentheses).

Number | Layout | Load |
---|---|---|

1 | PV | Inverter |

2 | PV | Start–Stop + Aggregated |

3 | PV | Inverter + Aggregated |

4 | PV + BES | Inverter |

5 | PV + BES | Start–Stop + Aggregated |

6 | PV + BES | Inverter + Aggregated |

7 | PV | Adaptive + Start–Stop |

8 | PV | Adaptive + Inverter |

9 | PV | Adaptive + Start–Stop + Aggregated |

10 | PV | Adaptive + Inverter + Aggregated |

11 | PV + BES | Adaptive + Start–Stop |

12 | PV + BES | Adaptive + Inverter |

13 | PV + BES | Adaptive + Start–Stop + Aggregated |

14 | PV + BES | Adaptive + Inverter + Aggregated |

Parameters | Values |
---|---|

Total installed PV capacity, P_{E} | $(1\u201320,\mathrm{step}0.05)\times $ P_{HP_Rated}, W |

BES Capacity, E_{B}^{inst} | $(0\u20135,\mathrm{step}0.1)\times $${P}_{\mathrm{E}},\mathrm{W}\times $ h |

Number of HPs | 1–4, step 1 |

${\mathit{f}}_{\mathit{min}}$, Hz | ${\mathit{f}}_{\mathit{max}}$, Hz | ${\mathit{f}}_{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$, Hz | ${\mathit{P}}_{\mathit{H}\mathit{P}\_\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$, kWe | Lifetime, Year | Cost, USD |
---|---|---|---|---|---|

30 | 70 | 50 | 6.5 | 10 | 4000 |

${\mathit{a}}_{\mathit{K}}$ | ${\mathit{b}}_{\mathit{K}}$ | ${\mathit{c}}_{\mathit{K}}$ | ${\mathit{d}}_{\mathit{K}}$ | Cost, USD/(A × h) |
---|---|---|---|---|

0 | 5714 | −14,571 | 11,857 | 2.75 |

${\mathit{a}}_{\mathit{K}\mathit{T}}$ | ${\mathit{b}}_{\mathit{K}\mathit{T}}$ | ${\mathit{c}}_{\mathit{K}\mathit{T}}$ | ${\mathit{d}}_{\mathit{K}\mathit{T}}$ | ${\mathit{e}}_{\mathit{K}\mathit{T}}$ | ${\mathit{f}}_{\mathit{K}\mathit{T}}$ | ${\mathit{g}}_{\mathit{K}\mathit{T}}$ |
---|---|---|---|---|---|---|

−3.218 × 10^{−10} | 4.323 × 10^{−8} | −1.348 × 10^{−6} | −2.906 × 10^{−5} | 8.922 × 10^{−4} | 2.740 × 10^{−2} | 3.624 × 10^{−1} |

${\mathit{T}}_{\mathit{r}\mathit{e}\mathit{f}}$, 1/°C | ${\mathit{K}}_{\mathit{T}}$, 1/°C | Lifetime, Year | Cost, USD/kW |
---|---|---|---|

38.8 | 2.85 × 10^{−3} | 30 | 700 |

RPS | Adaptive | Classic | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

CAPEX, UDD | P_{E}, kW | E_{B}^{inst}, kW × h | Control | HP Count | CAPEX, USD | P_{E}, kW | E_{B}^{inst}, kW × h | Control | HP Count | |

0.7 | 24.1k | 27.950 | 0 | Inverter | 2 | 27.0k | 26.0 | 13.00 | Inverter | 3 |

0.75 | 25.7k | 30.225 | 0 | Inverter | 2 | 33.4k | 33.8 | 16.90 | Inverter | 3 |

0.8 | 27.7k | 32.825 | 0 | Inverter | 2 | 39.7k | 36.4 | 36.40 | Inverter | 3 |

0.85 | 29.5k | 29.900 | 14.95 | Inverter | 2 | 39.7k | 42.9 | 21.45 | Start–Stop | 2 |

0.9 | 31.4k | 32.500 | 16.25 | Inverter | 2 | 42.1k | 40.3 | 40.30 | Start–Stop | 2 |

0.95 | 33.6k | 35.100 | 17.55 | Inverter | 2 | 46.3k | 44.2 | 44.20 | Start–Stop | 2 |

1 | 36.3k | 39.000 | 19.50 | Inverter | 2 | 50.5k | 49.4 | 49.40 | Start–Stop | 2 |

RPS | Adaptive | Classic | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

TCO, USD | P_{E}, kW | E_{B}^{inst}, kW × h | Control | HP Count | TCO, USD | P_{E}, kW | E_{B}^{inst}, kW × h | Control | HP Count | |

0.7 | 33.7k | 27.950 | 0 | Inverter | 2 | 46.0k | 30.550 | 6.110 | Inverter | 3 |

0.75 | 35.4k | 30.225 | 0 | Inverter | 2 | 52.0k | 46.475 | 4.648 | Start–Stop | 2 |

0.8 | 37.3k | 32.825 | 0 | Inverter | 2 | 57.4k | 54.600 | 5.460 | Start–Stop | 2 |

0.85 | 39.5k | 36.075 | 0 | Inverter | 2 | 64.1k | 53.625 | 10.725 | Start–Stop | 2 |

0.9 | 41.8k | 39.325 | 0 | Inverter | 2 | 71.5k | 61.750 | 12.350 | Start–Stop | 2 |

0.95 | 44.2k | 42.900 | 0 | Inverter | 2 | 80.5k | 51.025 | 30.615 | Start–Stop | 2 |

1 | 47.4k | 46.800 | 0 | Inverter | 2 | 85.4k | 73.125 | 21.940 | Start–Stop | 2 |

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## Share and Cite

**MDPI and ACS Style**

Klokov, A.V.; Tutunin, A.S.; Sharaborova, E.S.; Korshunov, A.A.; Loktionov, E.Y.
Inverter Heat Pumps as a Variable Load for Off-Grid Solar-Powered Systems. *Energies* **2023**, *16*, 5987.
https://doi.org/10.3390/en16165987

**AMA Style**

Klokov AV, Tutunin AS, Sharaborova ES, Korshunov AA, Loktionov EY.
Inverter Heat Pumps as a Variable Load for Off-Grid Solar-Powered Systems. *Energies*. 2023; 16(16):5987.
https://doi.org/10.3390/en16165987

**Chicago/Turabian Style**

Klokov, Alexander V., Alexander S. Tutunin, Elizaveta S. Sharaborova, Aleksei A. Korshunov, and Egor Y. Loktionov.
2023. "Inverter Heat Pumps as a Variable Load for Off-Grid Solar-Powered Systems" *Energies* 16, no. 16: 5987.
https://doi.org/10.3390/en16165987